There are 3 types of coupons each number [1,2,3] in a lottery. When you buy one lot you get only one coupon. What is the expected number of lots you need to buys to have at least one coupon of each number ?
Solution
step 1. At first you can collect one coupon with any number.
\[p (any \ number) = 1\]
Excpected number of lots to buy to get any one number = 1
step 2. Second number will be any number out of two numbers (first number has been collected in step 1)
\[p (step \ 2) = \frac{2}{3}\]
Excpected number of lots to buy to get second number ( \(\frac{1}{p} ) = \frac{3}{2} \)
Step 3: Collect third number
\[p(step \ 3) = \frac{1}{3}\]
Excpected number of lots to buy to get third number = 3
Excpected number of lots to buy to collect coupons of each of type = \(1 + \frac{3}{2} + 3\)
\(=3 * (1/3 + 1/2 + 1) = 5.5\)